Baron & Kenny (1986): Limitations

How much of a reduction in the relationship between the predictor and outcome is necessary to infer mediation?
People tend to look for a change in significance, which can lead to the ‘all or nothing’ thinking that p-values encourage.
Two solutions:
- The Sobel test
- Bootstrapping the indirect effect and confidence intervals

Sobel Test

An alternative is to estimate the indirect effect and its significance using the Sobel test (Sobel, 1982).
If the Sobel test is significant, there is significant mediation
We are going to use the Aroian Test version of Sobel which corrects for the standard error of the a and b paths.

Bootstrapping

What is bootstrapping?
- A procedure where you run an analysis many times to create an average effect and confidence interval of the effect.
- Bootstrapped datasets are created by randomly picking a sample from your dataset with replacement
- Think about this like the lottery, if they put the balls back.
- If the confidence interval of the indirect (mediation) effect does not cross zero, then that implies that mediation has occurred.

Mediation: Data Screening

Mediation models usually occur in stages - meaning separate models are run with different IV / DV combinations.
Sometimes people call these “steps”, but do not confuse this with hierarchical regression.
So, what does that do for data screening?
- Screen both the IV and the Mediator at the same time predicting the DV.
- This way you are screening the final anticipated stage of the mediation model and are screening all the variables.
- You can also use the fake regression procedure to do all three variables at once.

Mediation: Example

Does Facebook use mediate the relationship between previous knowledge and exam scores?
Implying that the overuse of Facebook prevents people from studying, so they do differently on their exam?
Important not to have missing data, so you are comparing models that all use the same data points.

library(rio)
master <- import("data/mediation.sav")
str(master)
#> 'data.frame':    240 obs. of  3 variables:
#>  $ previous: num  3 2 1 1 1 1 1 1 1 1 ...
#>   ..- attr(*, "format.spss")= chr "F1.0"
#>   ..- attr(*, "display_width")= int 14
#>  $ facebook: num  3.67 5 4 4.5 4.5 ...
#>   ..- attr(*, "format.spss")= chr "F8.2"
#>   ..- attr(*, "display_width")= int 14
#>  $ exam    : num  1 1 2 7 6 1 2 1 1 1 ...
#>   ..- attr(*, "format.spss")= chr "F8.2"
#>   ..- attr(*, "display_width")= int 10

summary(master)
#>     previous        facebook          exam       
#>  Min.   :1.000   Min.   :1.750   Min.   : 1.000  
#>  1st Qu.:1.000   1st Qu.:3.750   1st Qu.: 1.000  
#>  Median :1.000   Median :4.250   Median : 1.000  
#>  Mean   :1.571   Mean   :4.143   Mean   : 1.825  
#>  3rd Qu.:2.000   3rd Qu.:4.750   3rd Qu.: 2.000  
#>  Max.   :7.000   Max.   :5.000   Max.   :11.000  
#>                  NA's   :1
master <- na.omit(master)

Mediation: c Path

The c path: X –> Y: \[b = 0.32, t(237) = 3.63, p < .001\]

model1 <- lm(exam ~ previous, data = master)
summary(model1)
#> 
#> Call:
#> lm(formula = exam ~ previous, data = master)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -1.9092 -0.6477 -0.6477  0.3523  8.4062 
#> 
#> Coefficients:
#>             Estimate Std. Error t value
#> (Intercept)  1.33228    0.16749   7.955
#> previous     0.31539    0.08689   3.630
#>                       Pr(>|t|)    
#> (Intercept) 0.0000000000000737 ***
#> previous              0.000348 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.496 on 237 degrees of freedom
#> Multiple R-squared:  0.05266,    Adjusted R-squared:  0.04867 
#> F-statistic: 13.17 on 1 and 237 DF,  p-value: 0.0003476

Mediation: a Path

The a path: X –> M: \[b = -0.09, t(237) = -2.17, p = .031\]

model2 <- lm(facebook ~ previous, data = master)
summary(model2)
#> 
#> Call:
#> lm(formula = facebook ~ previous, data = master)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -2.44609 -0.44609  0.05391  0.55391  1.35640 
#> 
#> Coefficients:
#>             Estimate Std. Error t value
#> (Intercept)  4.28817    0.08195  52.330
#> previous    -0.09208    0.04251  -2.166
#>                        Pr(>|t|)    
#> (Intercept) <0.0000000000000002 ***
#> previous                 0.0313 *  
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.732 on 237 degrees of freedom
#> Multiple R-squared:  0.01941,    Adjusted R-squared:  0.01527 
#> F-statistic: 4.692 on 1 and 237 DF,  p-value: 0.03131

Mediation: b, c’ Path

Add in the b and c’ paths: X + M –> Y
b Path: \[b = -0.61, t(237) = -4.77, p < .001\]
c’ Path: \[b = 0.26, t(237) = 3.09, p = .002\]

model3 <- lm(exam ~ previous + facebook, data = master)
summary(model3)
#> 
#> Call:
#> lm(formula = exam ~ previous + facebook, data = master)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -2.6065 -0.7666 -0.3117  0.3850  7.9999 
#> 
#> Coefficients:
#>             Estimate Std. Error t value        Pr(>|t|)
#> (Intercept)  3.93294    0.56791   6.925 0.0000000000409
#> previous     0.25954    0.08397   3.091         0.00224
#> facebook    -0.60647    0.12705  -4.773 0.0000031782120
#>                
#> (Intercept) ***
#> previous    ** 
#> facebook    ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.432 on 236 degrees of freedom
#> Multiple R-squared:  0.1361, Adjusted R-squared:  0.1288 
#> F-statistic: 18.59 on 2 and 236 DF,  p-value: 0.00000003194

Mediation: Interpretation

Previous experience positively impacts exam scores (c path = 0.32).
Previous experience negatively impacts Facebook time (a path = -0.09).
Controlling for previous experience, Facebook time negatively impacts exam scores (b path = -0.61).
Controlling for Facebook scores, previous experience positively impacts exam scores (c’ path = 0.26).
So, did mediation happen? Is a change from 0.31 to 0.26 important?

Mediation: Sobel Test

The Aroian Test: \(Z = \frac{a \times b}{\sqrt{b^2 \times SD_a^2 + a^2 \times SD_b^2 + SD_a^2 \times SD_b^2}}\)
If the indirect effect is larger than the error, we would conclude that the addition of the M variable changed the c path.
- Fill in the correct coefficient numbers
- Note if you use this code provided and fill in X and M in the correct order, you will not need to change the Sobel code.

#aroian sobel
a <- coef(model2)[2]
b <- coef(model3)[3]
SEa <- summary(model2)$coefficients[2,2]
SEb <- summary(model3)$coefficients[3,2]
zscore <- (a*b)/(sqrt((b^2*SEa^2)+(a^2*SEb^2)+(SEa^2*SEb^2)))
zscore
#> previous 
#> 1.937494
#two tailed test 
pnorm(abs(zscore), lower.tail = F)*2
#>   previous 
#> 0.05268497

Mediation: Sobel Test

Z = 1.93, p = .053
We would conclude that no mediation had occurred.

Mediation: All Together + Bootstrapping

#devtools::install_github("doomlab/MeMoBootR")
library(MeMoBootR)
#> Loading required package: diagram
#> Loading required package: shape
#> 
#> Attaching package: 'shape'
#> The following object is masked from 'package:corrplot':
#> 
#>     colorlegend

#no missing data allowed
med_results <- mediation1(y = "exam",
                          x = "previous", 
                          m = "facebook", 
                          df = master)

Mediation: MeMoBootR

The MeMoBootR package gives you data screening, each step of the mediation, and the bootstrapping results!
The data screening does not include accuracy or missing data, so that should be completed first.

head(med_results$datascreening$fulldata)
#>   previous facebook exam badmahal badleverage badcooks
#> 1        3 3.666667    1        0           0        0
#> 2        2 5.000000    1        0           0        0
#> 3        1 4.000000    2        0           0        0
#> 4        1 4.500000    7        0           0        1
#> 5        1 4.500000    6        0           0        1
#> 6        1 4.500000    1        0           0        0
#>   totalout
#> 1        0
#> 2        0
#> 3        0
#> 4        1
#> 5        1
#> 6        0
med_results$datascreening$correl
#>             (Intercept)   previous   facebook
#> (Intercept)   1.0000000 -0.3617623 -0.9593469
#> previous     -0.3617623  1.0000000  0.1393247
#> facebook     -0.9593469  0.1393247  1.0000000
med_results$datascreening$linearity

med_results$datascreening$normality

med_results$datascreening$homogen

Mediation: MeMoBootR

For each of our stages of mediation, you can print out the models:

summary(med_results$model1)
#> 
#> Call:
#> lm(formula = allformulas$eq1, data = finaldata)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -1.9092 -0.6477 -0.6477  0.3523  8.4062 
#> 
#> Coefficients:
#>             Estimate Std. Error t value
#> (Intercept)  1.33228    0.16749   7.955
#> previous     0.31539    0.08689   3.630
#>                       Pr(>|t|)    
#> (Intercept) 0.0000000000000737 ***
#> previous              0.000348 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.496 on 237 degrees of freedom
#> Multiple R-squared:  0.05266,    Adjusted R-squared:  0.04867 
#> F-statistic: 13.17 on 1 and 237 DF,  p-value: 0.0003476
summary(med_results$model2)
#> 
#> Call:
#> lm(formula = allformulas$eq2, data = finaldata)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -2.44609 -0.44609  0.05391  0.55391  1.35640 
#> 
#> Coefficients:
#>             Estimate Std. Error t value
#> (Intercept)  4.28817    0.08195  52.330
#> previous    -0.09208    0.04251  -2.166
#>                        Pr(>|t|)    
#> (Intercept) <0.0000000000000002 ***
#> previous                 0.0313 *  
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.732 on 237 degrees of freedom
#> Multiple R-squared:  0.01941,    Adjusted R-squared:  0.01527 
#> F-statistic: 4.692 on 1 and 237 DF,  p-value: 0.03131
summary(med_results$model3)
#> 
#> Call:
#> lm(formula = allformulas$eq3, data = finaldata)
#> 
#> Residuals:
#>     Min      1Q  Median      3Q     Max 
#> -2.6065 -0.7666 -0.3117  0.3850  7.9999 
#> 
#> Coefficients:
#>             Estimate Std. Error t value        Pr(>|t|)
#> (Intercept)  3.93294    0.56791   6.925 0.0000000000409
#> previous     0.25954    0.08397   3.091         0.00224
#> facebook    -0.60647    0.12705  -4.773 0.0000031782120
#>                
#> (Intercept) ***
#> previous    ** 
#> facebook    ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 1.432 on 236 degrees of freedom
#> Multiple R-squared:  0.1361, Adjusted R-squared:  0.1288 
#> F-statistic: 18.59 on 2 and 236 DF,  p-value: 0.00000003194

Mediation: MeMoBootR

Next, you can get the Sobel test results:

med_results$indirect.effect
#>   previous 
#> 0.05584541
med_results$z.score
#> previous 
#> 1.937494
med_results$p.value
#>   previous 
#> 0.05268497

Mediation: MeMoBootR

Last, let’s get the bootstrapped results:
- The indirect effect would be reported as: \(0.06, 95\% CI[-0.01, 0.12]\)

med_results$boot.results
#> 
#> ORDINARY NONPARAMETRIC BOOTSTRAP
#> 
#> 
#> Call:
#> boot(data = finaldata, statistic = indirectmed, R = nboot, formula2 = allformulas$eq2, 
#>     formula3 = allformulas$eq3, x = x, med.var = m)
#> 
#> 
#> Bootstrap Statistics :
#>       original      bias    std. error
#> t1* 0.05584541 0.002617188  0.03151453
med_results$boot.ci
#> BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
#> Based on 1000 bootstrap replicates
#> 
#> CALL : 
#> boot.ci(boot.out = bootresults, conf = conf_level, type = "norm")
#> 
#> Intervals : 
#> Level      Normal        
#> 95%   (-0.0085,  0.1150 )  
#> Calculations and Intervals on Original Scale
med_results$diagram

Moderation

The combined effect of two variables on another is known conceptually as moderation, and in statistical terms as an interaction effect.
Do violent video games make people aggressive?
- DV: Aggression
- IV:
  - Callous unemotional traits
  - Number of hours spent playing video games per week
  - Interaction

Conceptual Moderation Model

If callous-unemotional traits were a moderator then we’re saying that the strength or direction of the relationship between game playing and aggression depends on the strength of callous-unemotional traits.

Conceptual Moderation Model

If the X variable was categorical, you are basically checking if the separate groups (levels) have different slopes for the non-categorical variable.

Conceptual Moderation Model

However, if X is continuous, we have to find a new way to figure out the shape of the interaction.

Statistical Moderation Model

Centering variables

The interaction term makes the bs for the main predictors uninterpretable in many situations.
Centering refers to the process of transforming a variable into deviations around a fixed point.
Centering solves two problems:
- Interpretation
- Multicollinearity

Centering variables

Easiest way to center to make them useful:
- Score - Mean for that variable.
- What does that do?
Centered variables for main effects have two interpretations
- Effect of that predictor at the mean value for the sample
- Average effect of the predictor across a range of scores for the other predictor

Moderation: Simple Slopes

If the interaction is significant, then you usually ignore the main effects (each variable by itself).
So what do I do if my interaction is significant? A simple slope analysis
If variables are categorical, then you are simply looking at the differences across one variable or another

Moderation: Simple Slopes

For continuous variables, you “create” low, average, and high groups.
- Low groups are people who are one SD below the mean
- Average groups are people are at the mean
- High groups are people who are one SD above the mean
- However, creating groups is backwards from what you think.

Moderation: Simple Slopes

We are examining the interaction between hours of video games and unemotional traits to predict aggression
Think about which variable you want to know the differences in (i.e., low, average, high).
So at different levels of callousness, we want to examine the relationship between hours of video games and aggression.

master <- import("data/moderation.sav")
str(master)
#> 'data.frame':    442 obs. of  4 variables:
#>  $ ID        : num  69 55 7 96 130 124 72 139 102 179 ...
#>   ..- attr(*, "label")= chr "Case ID"
#>   ..- attr(*, "format.spss")= chr "F3.0"
#>  $ Aggression: num  13 38 30 23 25 46 41 22 35 23 ...
#>   ..- attr(*, "label")= chr "Agression"
#>   ..- attr(*, "format.spss")= chr "F2.0"
#>  $ Vid_Games : num  16 12 32 10 11 29 23 15 20 20 ...
#>   ..- attr(*, "label")= chr "Video Games (Hours per week)"
#>   ..- attr(*, "format.spss")= chr "F2.0"
#>  $ CaUnTs    : num  0 0 0 1 1 1 2 3 3 3 ...
#>   ..- attr(*, "label")= chr "Callous Unemotional Traits"
#>   ..- attr(*, "format.spss")= chr "F2.0"

Moderation: Centering

Center your variables using scale()
- Using scale = F turns off dividing by SD, which would create z-scores.
- You can use z-score centering if you wanted the coefficients to be beta. It is likely easier to interpret the b values though.

#center the X and M variable (NOT THE DV)
#when you create these use the final dataset 
master$zvid = scale(master$Vid_Games, scale = F) #mean center, not z score
master$zcal = scale(master$CaUnTs, scale = F)

Moderation: Running

For data screening, we would screen the model with the interaction, and all other considerations are the same as regression.
Run the model:
- You do not have to create an interaction column, to multiply does that automatically for you.
- Also includes the main effects of each variable by itself.

#run the model to see if the moderation is significant
#use the z score variables!
#be sure to do X*M so the graph is right 
modmodel <- lm(Aggression ~ zvid*zcal, data = master)

Moderation: Interpretation

Overall model is significant, \(F(3, 438) = 88.46, p < .001, R^2 = .38\)
Interaction effect is significant, \(b = 0.03, t(438) = 3.88, p < .001\)

summary(modmodel)
#> 
#> Call:
#> lm(formula = Aggression ~ zvid * zcal, data = master)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -29.7144  -6.9087  -0.1923   6.9036  29.2290 
#> 
#> Coefficients:
#>              Estimate Std. Error t value
#> (Intercept) 39.967108   0.475057  84.131
#> zvid         0.169625   0.068473   2.477
#> zcal         0.760093   0.049458  15.368
#> zvid:zcal    0.027062   0.006981   3.877
#>                         Pr(>|t|)    
#> (Intercept) < 0.0000000000000002 ***
#> zvid                    0.013616 *  
#> zcal        < 0.0000000000000002 ***
#> zvid:zcal               0.000122 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 9.976 on 438 degrees of freedom
#> Multiple R-squared:  0.3773, Adjusted R-squared:  0.373 
#> F-statistic: 88.46 on 3 and 438 DF,  p-value: < 0.00000000000000022

Moderation: Simple Slopes

So, how do we create groups when we have a continuous variable?
- Most people look at “low” as one SD below the mean and “high” as one SD above the mean.
- Lots of combinations are possible though.

Moderation: Simple Slopes

Create the low and high versions of the M variable (callousness in this example)
- Low created by ADDING a SD
- High created by SUBTRACTING a SD
- The rule is that we have to bring them to the middle because we centered so that zero is the middle

#create the low and high z score variables 
master$zcallow <- master$zcal + sd(master$zcal) #bring them up
master$zcalhigh <- master$zcal - sd(master$zcal) #bring them down
summary(master)
#>        ID          Aggression      Vid_Games    
#>  Min.   :  1.0   Min.   : 9.00   Min.   : 1.00  
#>  1st Qu.:111.2   1st Qu.:32.00   1st Qu.:17.00  
#>  Median :221.5   Median :40.00   Median :22.00  
#>  Mean   :221.5   Mean   :40.05   Mean   :21.84  
#>  3rd Qu.:331.8   3rd Qu.:48.00   3rd Qu.:26.00  
#>  Max.   :442.0   Max.   :82.00   Max.   :38.00  
#>      CaUnTs           zvid.V1       
#>  Min.   : 0.0   Min.   :-20.843891  
#>  1st Qu.:11.0   1st Qu.: -4.843891  
#>  Median :18.0   Median :  0.156109  
#>  Mean   :18.6   Mean   :  0.000000  
#>  3rd Qu.:26.0   3rd Qu.:  4.156109  
#>  Max.   :43.0   Max.   : 16.156109  
#>        zcal.V1            zcallow.V1    
#>  Min.   :-18.595023   Min.   :-8.97733  
#>  1st Qu.: -7.595023   1st Qu.: 2.02267  
#>  Median : -0.595023   Median : 9.02267  
#>  Mean   :  0.000000   Mean   : 9.61769  
#>  3rd Qu.:  7.404977   3rd Qu.:17.02267  
#>  Max.   : 24.404977   Max.   :34.02267  
#>      zcalhigh.V1     
#>  Min.   :-28.212716  
#>  1st Qu.:-17.212716  
#>  Median :-10.212716  
#>  Mean   : -9.617693  
#>  3rd Qu.: -2.212716  
#>  Max.   : 14.787284

Moderation: Simple Slopes

Then run the model with the new low and high versions of the variable.
What do I do with that?
You look at the other variable because the coefficients for M and X*M will not change.

#run the models
#be sure to X*M
modmodellow <- lm(Aggression ~ zvid*zcallow, data = master)
modmodelhigh <- lm(Aggression ~ zvid*zcalhigh, data = master)

Moderation: Simple Slopes

At low levels of callousness, video game hours is unrelated to aggression, b = -0.09 …

summary(modmodellow) #low slope
#> 
#> Call:
#> lm(formula = Aggression ~ zvid * zcallow, data = master)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -29.7144  -6.9087  -0.1923   6.9036  29.2290 
#> 
#> Coefficients:
#>               Estimate Std. Error t value
#> (Intercept)  32.656771   0.672018  48.595
#> zvid         -0.090651   0.099105  -0.915
#> zcallow       0.760093   0.049458  15.368
#> zvid:zcallow  0.027062   0.006981   3.877
#>                          Pr(>|t|)    
#> (Intercept)  < 0.0000000000000002 ***
#> zvid                     0.360854    
#> zcallow      < 0.0000000000000002 ***
#> zvid:zcallow             0.000122 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 9.976 on 438 degrees of freedom
#> Multiple R-squared:  0.3773, Adjusted R-squared:  0.373 
#> F-statistic: 88.46 on 3 and 438 DF,  p-value: < 0.00000000000000022

Moderation: Simple Slopes

At average levels of callousness, hours of video games predicts increasing aggression, b = 0.17 …

summary(modmodel) #average slope
#> 
#> Call:
#> lm(formula = Aggression ~ zvid * zcal, data = master)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -29.7144  -6.9087  -0.1923   6.9036  29.2290 
#> 
#> Coefficients:
#>              Estimate Std. Error t value
#> (Intercept) 39.967108   0.475057  84.131
#> zvid         0.169625   0.068473   2.477
#> zcal         0.760093   0.049458  15.368
#> zvid:zcal    0.027062   0.006981   3.877
#>                         Pr(>|t|)    
#> (Intercept) < 0.0000000000000002 ***
#> zvid                    0.013616 *  
#> zcal        < 0.0000000000000002 ***
#> zvid:zcal               0.000122 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 9.976 on 438 degrees of freedom
#> Multiple R-squared:  0.3773, Adjusted R-squared:  0.373 
#> F-statistic: 88.46 on 3 and 438 DF,  p-value: < 0.00000000000000022

Moderation: Simple Slopes

At high levels of callousness, the strength of hours of video games predicting aggression is the strongest, b = 0.43 …

summary(modmodelhigh) #high slope
#> 
#> Call:
#> lm(formula = Aggression ~ zvid * zcalhigh, data = master)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -29.7144  -6.9087  -0.1923   6.9036  29.2290 
#> 
#> Coefficients:
#>                Estimate Std. Error t value
#> (Intercept)   47.277446   0.672520  70.299
#> zvid           0.429901   0.092577   4.644
#> zcalhigh       0.760093   0.049458  15.368
#> zvid:zcalhigh  0.027062   0.006981   3.877
#>                           Pr(>|t|)    
#> (Intercept)   < 0.0000000000000002 ***
#> zvid                    0.00000453 ***
#> zcalhigh      < 0.0000000000000002 ***
#> zvid:zcalhigh             0.000122 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 9.976 on 438 degrees of freedom
#> Multiple R-squared:  0.3773, Adjusted R-squared:  0.373 
#> F-statistic: 88.46 on 3 and 438 DF,  p-value: < 0.00000000000000022

Moderation: Simple Slopes

Significant interactions will create fan effects of slopes that either get stronger/weaker in strength or switch directions.
Here, we find that increasing levels of callousness leads to increasing effects of video game time on aggression.

Moderation: Graphing

We can use the ggplot to create a graph of the moderation effect.
But we are going to manually draw the lines on a scatterplot graph.

library(ggplot2)
cleanup <- theme(panel.grid.major = element_blank(), 
                panel.grid.minor = element_blank(), 
                panel.background = element_blank(), 
                axis.line.x = element_line(color = "black"),
                axis.line.y = element_line(color = "black"),
                legend.key = element_rect(fill = "white"),
                text = element_text(size = 15))

modgraph <- ggplot(master, aes(zvid, Aggression))

##change Cal to the new moderator label
##change xlab for the new X label
modgraph + 
  xlab("Centered Video Games") + 
  geom_point(color = "gray") +
  geom_abline(aes(intercept = modmodellow$coefficients[1],
                  slope = modmodellow$coefficients[2], 
                  linetype = "-1SD Cal"), size = 1) +
  geom_abline(aes(intercept = modmodel$coefficients[1],
                  slope = modmodel$coefficients[2], 
                  linetype = "Average Cal"), size = 1) +
  geom_abline(aes(intercept = modmodelhigh$coefficients[1],
                  slope = modmodelhigh$coefficients[2], 
                  linetype = "+1SD Cal"), size = 1) +
  scale_linetype_manual(values = c("dotted", "dashed", "solid"),
                        breaks = c("-1SD Cal", "Average Cal", "+1SD Cal"),
                        name = "Simple Slope") +
  cleanup

Moderation: MeMoBootR

We can use the MeMoBootR to complete the entire processing, including data screening for us!
You would enter the raw variables, as the centering is completed for you.

mod_model <- moderation1(y = "Aggression",
                         x = "Vid_Games",
                         m = "CaUnTs",
                         df = master)

Moderation: MeMoBootR

#data screening
head(mod_model$datascreening$fulldata)
#>    ID Aggression Vid_Games CaUnTs       zvid      zcal
#> 1  69         13        16      0  -5.843891 -18.59502
#> 2  55         38        12      0  -9.843891 -18.59502
#> 3   7         30        32      0  10.156109 -18.59502
#> 4  96         23        10      1 -11.843891 -17.59502
#> 5 130         25        11      1 -10.843891 -17.59502
#> 6 124         46        29      1   7.156109 -17.59502
#>    zcallow  zcalhigh badmahal badleverage badcooks
#> 1 -8.97733 -28.21272        0           0        1
#> 2 -8.97733 -28.21272        0           1        0
#> 3 -8.97733 -28.21272        0           1        0
#> 4 -7.97733 -27.21272        0           1        0
#> 5 -7.97733 -27.21272        0           1        0
#> 6 -7.97733 -27.21272        0           1        1
#>   totalout
#> 1        1
#> 2        1
#> 3        1
#> 4        1
#> 5        1
#> 6        2
#mod_model$datascreening$correl
#mod_model$datascreening$linearity
#mod_model$datascreening$normality
#mod_model$datascreening$homogen

#models
summary(mod_model$model1)
#> 
#> Call:
#> lm(formula = allformulas$eq1, data = finaldata)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -29.7144  -6.9087  -0.1923   6.9036  29.2290 
#> 
#> Coefficients:
#>                   Estimate Std. Error t value
#> (Intercept)      39.967108   0.475057  84.131
#> Vid_Games         0.169625   0.068473   2.477
#> CaUnTs            0.760093   0.049458  15.368
#> Vid_Games:CaUnTs  0.027062   0.006981   3.877
#>                              Pr(>|t|)    
#> (Intercept)      < 0.0000000000000002 ***
#> Vid_Games                    0.013616 *  
#> CaUnTs           < 0.0000000000000002 ***
#> Vid_Games:CaUnTs             0.000122 ***
#> ---
#> Signif. codes:  
#> 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 9.976 on 438 degrees of freedom
#> Multiple R-squared:  0.3773, Adjusted R-squared:  0.373 
#> F-statistic: 88.46 on 3 and 438 DF,  p-value: < 0.00000000000000022
#summary(mod_model$model1low)
#summary(mod_model$model1high)
mod_model$interpretation
#> [1] "At low levels of CaUnTs, you see that every unit increase in Vid_Games predicts -0.09 unit change in Aggression. \n\nAt average levels of CaUnTs, you see that every unit increase in Vid_Games predicts 0.17 unit change in Aggression. \n\nAt high levels of CaUnTs, you see that every unit increase in Vid_Games predicts 0.43 unit change in Aggression."

#graphs
mod_model$graphslopes

Mediation and Moderation

What is Mediation?

Baron & Kenny (1986)

Baron & Kenny (1986)

Baron & Kenny (1986): Limitations

Sobel Test

Bootstrapping

Mediation: Data Screening

Mediation: Example

Mediation: c Path

Mediation: a Path

Mediation: b, c’ Path

Mediation: Interpretation

Mediation: Sobel Test

Mediation: Sobel Test

Mediation: All Together + Bootstrapping

Mediation: MeMoBootR

Mediation: MeMoBootR

Mediation: MeMoBootR

Mediation: MeMoBootR

Moderation

Conceptual Moderation Model

Conceptual Moderation Model

Conceptual Moderation Model

Statistical Moderation Model

Centering variables

Centering variables

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Centering

Moderation: Running

Moderation: Interpretation

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Simple Slopes

Moderation: Graphing

Moderation: MeMoBootR

Moderation: MeMoBootR

Summary